neutron powder diffraction study of order–disorder phase transit in cc

8/12/2019 Neutron Powder Diffraction Study of OrderDisorder Phase Transit in Cc

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O R IGINA L PA PE R

Martin T. Dove Ian P. Swainson Brian M. PowellDonald C. Tennant

Neutron powder diffraction study of the orientational orderdisorderphase transition in calcite, CaCO3

Received: 7 April 2005 / Accepted: 29 July 2005 / Published online: 27 September 2005 Springer-Verlag 2005

Abstract Neutron powder diffraction studies of calciteon heating towards the orientational orderdisorderphase transition show that the phase transition is not a

simple analogue of an Ising-like transition, but moresimilar to a rotational analogue of Lindemann melting.The transition is precipitated by the librational ampli-tude of the carbonate molecular ions exceeding a criticalvalue rather than a result of a statistical entropy ofwrong orientations. Using tested interatomic potentialsthe single-particle orientational potential and nearest-neighbour orientational interactions have been calcu-lated.

Introduction

The existence of a phase transition in calcite (CaCO3) athigh temperature has been suspected for a long time, buta detailed structural experimental investigation has notbeen possible because the carbonate group is chemicallyunstable at temperatures below the supposed transitiontemperature. Having an environment of CO2 gas at apressure of only a few atmospheres can retard thedecomposition of the carbonate group. Because of thehigh penetrating power of neutrons compared to X-rays,it is possible to use standard furnaces to enclose therequired environment. In a previous neutron diffraction

study (Dove and Powell 1989), the temperature

dependence of two superlattice Bragg peaks, 113 and211, was monitored. It was observed that they bothdisappeared atTc1,260 K, consistent with a halving of

thec-lattice parameter and a change in the space groupfromR3ctoR3m:The intensity data were consistent witha temperature dependence of an order parameter Q,which was not characterized, of the form:

Q / I1=2lodd/ Tc T

1=4; 1

whereIl=oddis the intensity of a Bragg peak with an oddvalue of l.

The temperature dependence of the Bragg angles forthe few reflections that were measured suggested theexistence of a large spontaneous strain e3 that by sym-metry should be proportional to Q2. The strain datawere accurately described by the function:

e3 / Tc T 1=2: 2

Thus both the intensity data and the unit cell dataappeared consistent with the temperature dependence ofthe order parameter of the form Q / Tc T

1=4; whichis for a classical tricritical phase transition (Bruce andCowley 1981). Usually this follows from a Landau freeenergy function GL of the form

GL 1

2g2Q

2 1

6g6Q

6; 3

which can arise accidentally from a Landau free en-

ergy with a strain-coupling term that in this case couplesthe large e 3 strain to Q

2:

GL 1

2g2Q

2 1

4g4Q

4 1

6g6Q

6 1

2ae3Q

2 1

2C33e

23; 4

where C33 is an elastic constant. Minimization with re-spect to e 3 yields

e3 aQ2

2C335

M. T. Dove (&)Department of Earth Sciences, University of Cambridge,Downing Street, Cambridge, CB2 3EQ, UKE-mail: [email protected]

I. P. Swainson B. M. Powell D. C. Tennant M. T. DoveCanadian Neutron Beam Centre,National Research Council of Canada,Chalk River Laboratories, Chalk River,ON, K0J 1J0, Canada

Phys Chem Minerals (2005) 32: 493503DOI 10.1007/s00269-005-0026-1


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and

GL 1

2g2Q

2 1

4 g4

a2

2C33

Q4

1

6g6Q

6: 6

Thus, (6) reduces to (3) when

g4 a2

2C33: 7

The experimental findings for calcite are similar tothe behaviour observed in the analogous phase transi-tion in sodium nitrate, NaNO3 (Reeder et al. 1988;Schmahl and Salje 1990; Swainson et al. 1998). Thelatter is simpler to study as it has a lower Tc and doesnot require gas pressure to prevent decomposition, al-though the melting point lies not far above Tc.

It is unlikely that in the cases of both calcite andsodium nitrateb1/4 will be due to a cancellation of thefourth-order term in the Landau free energy expansionas expressed by (7). A tricritical point at ambient pres-sure would be an accident and the likelihood of bothcompounds having a tricritical point at the same pres-

sure seems very small. In the case of NaNO3 the expo-nent on the order parameter in (2) has been reported asdropping from 0.25 to 0.22 close to the transition tem-perature (Reeder et al. 1988; Schmahl and Salje 1990).

More recently, Harris (1999) argued strongly againsttricriticality as an explanation of b1/4 in either tran-sition. He proposed that both transitions could bemodelled by a two-dimensional XY pseudospin model,which predicts values ofb close to 0.23. Such a modelimplies that the disordering process is two-dimensional.This model appears to fit the static structure of calcite inthe ordered phase, where the planar carbonate ions lie in001 planes, and assumes insignificant coupling between

layers. He also questioned the existence of a cross-overofb from 0.25 to 0.22 in NaNO3, arguing that a value of0.22 appeared to be valid over the entire temperaturerange until about 5.5 K ofTc, at which point the fluc-tuations were so large that the interactions becomethree-dimensional and b values of approximately 0.34were more appropriate. We cannot settle the debate onthe exact value ofb from the data presented here, but wewill comment on the validity of pseudospin models. Inparticular, our evidence suggests that there is a strongcoupling of carbonate ions between the planes, revealedby large out-of-plane motion of these ions on heating.

There have been inelastic neutron scattering mea-

surements (Dove et al. 1992a; Hagen et al. 1992; Harriset al. 1998) and computational studies on calcite (Doveet al. 1992b; Ferrario et al. 1994; Liu et al. 2001). Theinelastic neutron scattering study showed the existenceof an unexpected feature, namely strong inelastic scat-tering localized at a few points on the surface of theBrillouin zone (corresponding to the F-points on thesurface of the Brillouin zone of the disordered phase)and which extended up to about 2 THz in energy. Dueto the narrow width of this feature in reciprocal space, ithas been described as a column of scattering. This col-

umn appears to be associated with a softening of one ofthe phonon branches at this pointa branch that is thetransverse acoustic mode at small wave vectors. Onheating, the intensity of the column increases dramati-cally, the frequency of the mode decreases and the modebecomes strongly damped. But within the resolution ofthe experiment, the width of the column does notbroaden in Q on heating, reflecting some kind of long-range ordering (Harris et al. 1998). Conversely it dis-appears completely on cooling towards 0 K (Dove et al.1992a).

In order to provide some insight into the inelasticbehaviour a simple interatomic potential model has beendeveloped for calcite (Dove et al. 1992b). This was usedin static lattice energy relaxation calculations and incalculations of the harmonic lattice dynamics, whichreproduced the softening of the transverse acousticbranch as described above. The model was used in anMD simulation of calcite (Ferrario et al. 1994), andmany of the features of this simulation have been re-cently confirmed by more modern MD methods (Liuet al. 2001). The simulations reproduce many of the

experimental observations remarkably well, e.g. thetransition temperature, the large spontaneous straine

3and the results showing the growth of the librationalamplitude, L33, of the carbonate group that are pre-sented in this paper.

The simulation of Ferrario et al. (1994) showed thatthe symmetry of the phonon branch that is the trans-verse acoustic branch at smallk is of symmetryF2 .F2

is also the symmetry of the soft mode for the lowesthigh-pressure transition in calcite, to calcite II (Merrilland Bassett1975; Hatch and Merrill1981). On the otherhand, the simulations have been unable to reproduce thecolumnar nature of the observed inelastic scattering. The

narrow width of the column in reciprocal space suggeststhat theF-point ordering must involve correlations overdistances that are much larger than the size of thesample used in molecular dynamics simulation.

The column is a rather unusual feature in a set ofphonon dispersion curves. Its closest analogues appearto be b-Ti (Petry et al. 1988) and b-Zr (Heiming et al.1991), where a column in the dispersion descends verysteeply and whose eigenvectors also correspond to thex-phase of these metals, related to the b-phase by a dis-placive transition at pressure. In the case of calcite, thebehaviour could be fitted to an interaction between thephonon itself and a relaxational mode (Harris et al.

1998), using the model of Michel and Naudts (1978).This showed that the phonon frequency approaches zeroas Tc is approached, and the value of c, a correlationfunction representing coupling between phonon andrelaxational modes, appears to approach unity onheating towardsTc. A frequency mode approaching zeroand a value of c tending to one are both signs of animminent phase transition. It therefore appears that nearthe transition point for Z-point ordering (ca. 1,250 K)another nearly energetically degenerate ordering iscompeting at the F-point.

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In order to shed further light on the features of thisphase transition and the general mechanism of disor-dering, it is important to obtain good structural data athigh temperatures, ideally to temperatures above Tc.Markgraf and Reeder (1985) have previously performeda single-crystal X-ray study up to 1,073 K, which issome 200 K short of Tc, but chemical decompositionprevented studies at higher temperatures. We havetherefore attempted to extend the structural informationto higher temperatures using neutron scattering.

Experimental details

Our present experiments were performed on the C2DUALSPEC neutron powder diffractometer at theNRU reactor, Chalk River Laboratories. C2 has an 800-element detector covering an angular range of 80. Thewavelength selected was 1.50329(5) A , calibrated via anexternal standard of NBS Si 640b. We collected data ateach temperature for scattering angles between 5 and117. Two attempts were made to stabilize calcite into

the disordered phase, resulting in two datasets.Dataset 1. A sample of calcite powder was loaded

into an open vanadium can of diameter 5 mm, whichwas then mounted in a furnace. In order to preventchemical decomposition the sample was maintained inan overpressure of CO2 gas. However, at high tem-peratures CO2 is highly corrosive, and for this reasonspecial furnace elements were required. The heatingelement was a cylinder of SiC, which surrounded thesample in the scattering plane. SiC is resistant to oxi-dation since the initial oxidation produces a silicasurface layer that prevents further oxidation. Thethermocouples, in direct contact with the sample, were

sheathed with a coating of stainless steel to preventoxidation. The inside of the furnace was evacuated andthen filled with CO2 to a pressure of 2 atm as measuredby a gauge on the surface, which was held at ambienttemperature. The pressure at the sample position isexpected to be higher following ideal gas heating, butits value cannot be determined. Data were collected upto nominal temperatures of 1,155 K. Despite ourcareful preparations we found that we were unable toprevent decomposition on heating above 1,200 K.Pressurizing the entire furnace beyond 2 atm was notpossible.

Dataset 2. The experiment was then repeated in a

different furnace in which the calcite was placed into avanadium foil, which in turn was inside a large vana-dium tube. This tube can be pressurized up to 9 atm ofCO2 (as measured at room temperature) with the aid ofa water-cooled seal, leaving the body of the furnace invacuum. A full description of this equipment is found inRollings et al. (1998). The only additional experimentaldifficulty with this setup up was that a sample thermo-couple could not be placed in direct contact with thesample inside the pressure chamber. Therefore, themeasured temperature was in error. We used the c-lattice

parameter as an internal calibrant over the range oftemperatures that were measured in common. Using thiswe were able to reference the results from the latterfurnace to those of the former furnace, which had athermocouple in direct contact with the sample. A verylinear temperature offset was found between 500 and1,155 K (R2=0.9997).

Data analysis

The diffraction data were analysed using two methods,Rietveld refinement and extraction of peak intensities.Rietveld refinement is not particularly suited for thestudy of orientational disorder, but we expect that thephases generated for the individual structure factors willbe correct and therefore of use for producing Fouriermaps with the observed structure factors. For referencewe show the crystal structure of the ordered phase inFig.1. The structure consists of planar trigonal car-bonate groups coordinated to Ca2+ ions in distortedoctahedra. The carbonates lie in layers normal to [001].

The carbonate groups within a single layer have thesame orientation. In the neighbouring layers the car-bonate groups are rotated by 60 about [001]. Hence in

Fig. 1 The crystal structure of calcite in the hexagonal cell of thelow-temperature ordered phase. The carbonate groups are theplanar trigonal units and the calcium the isolated atoms. Thecarbonate groups lie in planes normal to c, which runs vertically.Within any plane all the carbonate groups point in the samedirection. Neighbouring planes of carbonate groups have oppositeorientations

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the ordered phase the layers of carbonate groups alter-nate in orientation. In the disordered phase there is nodistinction between the neighbouring layers, and for anycarbonate group both orientations (h=0 and h=60)are equally probable. This results in a halving of thec-axis in the disordered phase.

In Fig.2 we show the temperature dependence of theunit cell parameters. These are consistent with our pre-vious data and show the same small negative thermalexpansion of a and the large thermal expansion of

c associated with the spontaneous strain e3. Thec-parameter of the ordered R3c cell is twice that of theR3m cell of the disordered phase, when expressed inthe hexagonal setting. The large expansion of the c-lat-tice parameter is anomalous, since it is clearly coupled tothe phase transition, yet is normal to the plane in whichthe main disordering process of the carbonate groupstakes place. The origin of this effect is discussed below.Since we have used the complete diffraction pattern toobtain the cell parameters, rather than using only two

Fig. 2 Temperaturedependence of the unit cellparameters a and c for the twodatasets, as numbered in thetext. The a cell shows a smalloverall contraction on heating.The expansion of the c-latticeparameter is large. The esds forthe values ofc are smaller thanthe points. The linerepresents ahypothetical extension of the

c-parameter of the disorderedphase, assuming therelationship given by (2) in thetext

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reflections as in the study of Dove and Powell (1989), wehave significantly improved upon the precision of the

measurements. The straight line in Fig.2 represents theextrapolated value of twice the value of the c-latticeparameter of the disordered phase, using all the mea-sured values ofc and assuming that (2) holds, and thatexpansion in the disordered phase is reasonablyapproximately by a linear expansion. Our best estimatefrom these method yields 8.8438+6.5527103TA ,where T is in Kelvin. Due to the higher temperaturereached in this study and the fact that we use all theBragg peaks, we consider this to be a better estimatethan that given earlier (Dove and Powell 1989).

The small plateau region in the expansion of the a cellparameter at around 1,000 K was also seen in the pre-

vious neutron powder diffraction study (Dove andPowell1989). With the extension of the dataset to highertemperatures in this study it is shown that this plateau isfollowed by another region of contraction above ca.1,050 K.

The remaining part of the analysis below deals withthe more extensive second dataset. In the ordered phase,reflections hkl with l odd are superlattice reflectionsgenerated by the ordering of the carbonate groups. InFig.3 we show a stack plot of the diffraction patternsof calcite on heating up to 1,250 K. The lowest order

Fig. 3 Stack plot of diffractionpatterns ofR3c calcite onheating from room temperatureto 1,250 K. The highesttemperature diffraction patternshows some additional peaksdue to the beginning ofbreakdown of the sample

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superlattice peak 113 at 39 2h can be seen to disap-pear, at least to the level at which we could measurethem. This would be a signal that we had entered thedisordered phase, but it is always possible that we lackthe signal-to-noise ratio sufficient to claim this. The newpeaks that appear correspond to reflections from CaO,due to the onset of breakdown of the sample. Twoadditional diffraction patterns were taken at 1,260 and1,270 K, but it is difficult to display them as CaO be-comes increasingly dominant due to the serious break-down of the sample near the disordered phase. It wasstill possible to determine lattice parameters, but littleother information.

In Fig.4 we show the temperature dependence of theintensities of a number of reflections from the datapresented in Fig.3. The reflections with even values oflare not expected to disappear on heating into the high-temperature phase, and these show a temperaturedependence that is only determined by the general in-crease in temperature factors with temperature. There-fore the intensities of the lowest angle reflections witheven lbarely change with temperature. In contrast thereflections with odd values of l have intensities that de-crease on heating towards the transition temperature.The lower angle reflections with odd values of l areconsistent with (1) to within statistical uncertainty. The

higher angle data deviate slightly from the form of (1)because they are more sensitive to the increase in tem-perature factors, whose effects are also clearly seen in thetemperature dependence of the reflections with evenvalues of l. The square of the intensity of the lowestorder superlattice peak, 113, is shown as a function oftemperature in Fig.4, together with a best-fit straightline representing (1) in accordance with values ofb near1/4.

We attempted refinements of calcite using a classicalspin model of disorder, but such refinements were not

stable at any temperature, as reported by previousworkers in calcite and NaNO3 (e.g. Markgraf andReeder 1985; Gonschorek et al. 1995). The crystal

Fig. 4 Temperaturedependence of the intensities ofisolated reflections with botheven and odd values ofl. Indescending order these are 102,104, 006, 113, 211 and 21, 11.Odd lreflections all show astrong decrease in intensity onheating, as these are superlatticepeaks generated by the orderingof the carbonate groups below

Tc. Even lreflections show amuch smaller reduction inintensity and are only affectedby the general increase in allDebyeWaller factors. Allintensities have beennormalized to unity at roomtemperature.Inset The squareof the intensity of the lowestorder superlattice peak, 113, asa function of temperature

Fig. 5 Observed Fourier maps of calcite for a section viewed down[001] at 0 0 1/4 containing the carbonate ions, from the 1,189 Kdata. A schematic of a carbonate ion is placed on top of the majorobserved intensity group. The Ca ion is also labelled. The mapshows that the centre of mass of carbon, as shown in the centre ofthe figure, is still well defined on approaching Tc. The scale hasbeen deliberately strongly amplified. While this shows up somenoise, it was done so in order to show the extent of the spread oforientations of the carbonate groups. However, it is clear that thedistribution of orientations does not include a significant popula-tion of the anti-ordered orientations, i.e. those rotated by 60

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structure of calcite at high temperatures is best illus-trated using Fourier maps. In Fig. 5we show a Fouriermap with an exaggerated scale, calculated using theobserved structure factors and the calculated phasevalues from the data taken at 1,189 K. The most strikingfeature is the large librational amplitude of the carbon-ate groups for rotations about [001], which increasesdramatically on heating. There is also an absence of anydistinct nuclear density at positions that would corre-spond to oxygens being in the anti-ordered positions, i.e.with the carbonate ion having rotated by 60 about[001]. These observations follow the trends found byMarkgraf and Reeder (1985) at lower temperatures. If

the transition were due to a classical pseudospin model,assuming a value ofb0.25, we would expect approxi-mately 1/4 of the carbonates to be in the anti-orderedorientation in Fig. 5. A similar absence of any anti-ordered groups was seen in the combined synchrotronand neutron single-crystal studies performed on NaNO3by Gonschorek et al. (1995) at room temperature.

In order to quantify these features we have analysedthe atomic motions given by the atomic temperaturefactors associated with the carbonate groups in terms ofthe rigid body TLS tensors, the approach previouslyused by Markgraf and Reeder (1985) in the analysis ofX-ray single-crystal diffraction data of calcite. We used

T (K)

200 400 600 800 1000 1200 1400

RMSlibrationalamplitude(deg)

0

5

10

15

20

25

30

35

40

L11

1/2

L33

1/2

Temperature K

200 400 600 800 1000 1200 1400

Componentdisplacem

entA

2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

T11

T33

b2L11

2

b2

L332

Fig. 6 Temperaturedependence of the thermalparameters T11, T33, b

2L11andb2L33 (top), where b is the CObond length, and of thelibrational amplitudes L11andL33 (bottom)

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the TLS facility in both GSAS (Larson and VonDreele(1987) for the results presented here, although we havechecked them with refinements using Fullprof (Rodri-guez-Carvajal 1990) and no significant differences werefound. These serve as an illustration of the main disor-dering processes, as at high librational amplitude theapproximation that thermal motion is a small pertur-bation of a static structure becomes much poorer. Wedid not find the screw components to be stable undereither of these programs, so we present onlyTandL. In

Fig.6 we show the temperature dependence of thethermal parametersT11,T33,b

2L11and b2L33, whereb is

the CO bond length, T11 and T33 are rigid bodytranslations of the carbonate group along axes parallelto [100] and [001], respectively, and L11 and L33 are li-brations of the carbonate group about the same axes.L11andT33rotate and translate the carbonate group outof the (001) plane, whereas L33 and T11 correspond tothe in-plane motions. It is clear that the increase of theL33 librational components far exceeds that of all othercomponents. However, there is a substantial growth of

L11 corresponding to the out-of-plane motion of thecarbonate ions. This suggests that the spins are not de-coupled between neighbouring planes. It may also be theorigin of the large thermal expansion along c, since thereis an increase in the effective size of the carbonate ionalong [001] as the disorder increases.

In Fig.6 we also show the temperature dependence ofthe librational amplitudes L11 and L33the rapidincrease ofL33on heating is clearly seen. The growth of

L33 was also observed in the molecular dynamics sim-ulations of Ferrario et al. (1994) and Liu et al. (2001).Once the amplitude for librations about [001] reaches30 each carbonate group will be able to flip easily intothe anti-ordered orientation. In the disordered phasesfor calcite and NaNO3 both simulations suggest thatthere is some preferred occupation in anti-ordered sites,superimposed on a general background of orientationaldisorder. It appears that the large librational amplitudesnear the disordered phase are responsible for thebreakdown on heating of calcite and the relatively smalltemperature range of the disordered phase in NaNO3before the melting point. In this respect, it bears some

resemblance to the hexahydrate, ikaite, in which hin-dered librations of the carbonate ion have also beensuggested as the mechanism of decomposition of thatmetastable phase (Swainson and Hammond2003).

Discussion

We have observed a rapid decrease in intensity of thereflections with odd values of l on heating, consistentwith the change in space-group symmetry. The presenceof distinct hkl, l=2n+1, superlattice peaks in the com-paratively well-ordered state of calcite at room temper-

ature which are lost at high temperature only givesinformation that there is an alternating order along the[100] direction, which is destroyed on heating. By di-rectly examining the structure of calcite or by calculatingexplicitly the structure factors, it is clear that thisordering is orientational ordering of carbonate groups,as defined by oxygen atoms. Our structure analysis hasshown the rapid growth of thermal motion, in particularin the librational amplitude on heating towards thephase transition.

The disorder we envisage for the high-temperaturephase, and the only one that is consistent with thechange in space-group symmetry, is that each carbonate

group occupies a site of point symmetry 3m, so thatthere is an equal probability of locating an oxygen atomat intervals of 60. The simplest way to view this is tothink of each carbonate group as having two possibleorientations related to each other by rotations of 60(orequivalently 180). Thus one might assign a pseudospinto each site, of value 1 depending on whether theorientation is 0 or 60. We would then expect that atlow temperature all the carbonate groups in any givenplane will have the same orientations and on heating thenumber of carbonate groups in the same plane with the

Fig. 7 a Interaction potential between two nearest-neighbourcarbonate pairs in the same (001) plane as a function of theorientations of the two groups. The relative orientations fordifferent points on the contour map are shown as symbolic axislabels, alongside the angular values. Solid contours are inincrements of 1,000 K. The maximum at (60 , 0) has a relativeenergy of 8,376 K; the saddle point at (60, 60) has a relativeenergy of 0 K; and the minimum at (0, 60) has a relative energy of2,848 K. b Interaction potential between two nearest-neighbourcarbonate pairs in neighbouring (001) planes (separated by c/6) as afunction of the orientations of the two. Dashed contours are inintervals of 50 K. The maximum at (30, 90) has a relative energyof 348 K; the saddle point at (90, 30) has a relative energy of236 K; and the minima at (30, 30) and (90, 90) have relativeenergies of281 K

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opposite orientation will increase, similar to the behav-iour found in any Ising-like phase transition. The orderparameter would then be defined as the difference be-tween the number of carbonate molecular ions in thecorrect (ordered) and wrong (anti-ordered) orienta-tions, which are related by 60 rotations.

This pseudospin scenario, as we have seen in theprevious section, is not borne out by our Fourier mapsat high temperatures, which show no sign of any car-bonate groups in opposite orientations. This observationwas also previously reported from the X-ray results ofMarkgraf and Reeder (1985), for temperatures up to1,073 K. Instead of a simple statistical disorder, we findonly large-amplitude librations about [001], which onheating increase in size much more rapidly than theusual linear temperature dependence. This effect wasalso seen in the molecular dynamics simulation as abroadening of the single-molecule orientational distri-bution function for the carbonate groups (Ferrario et al.1994; Liu et al2001).

We are therefore forced to view the phase transitionin a new way. We start by considering the local potential

at the carbonate site due to the surrounding calciumatoms. In the disordered phase this has point symmetry3m, so the potential is the same for rotations of 60 andcan be described by a local single-particle potentialVLocal(J) of the form:

VLocal# V0 sin23#; 8

where h describes the orientation of the carbonate ionabout [001], and h=0 gives the average orientation inthe layer shown in Fig.5 (the values ofh alternate be-tween 0 and 60 on going between the planes). Thedescription in terms of pseudospins is equivalent to the

approximation that V0fi

. Using the potential ofDove et al. (1992b), we have calculated a value forV04,500 K, assuming that the most important contri-butions are the short-range interactions with the neigh-bouring Ca ions and the long-range Coulombinteractions. For the Coulomb interactions we havecalculated the potential for a single carbonate ion in afield due to point charges and have assumed that attypical distances the neighbouring carbonate groups canbe represented as point charges at the carbon position.

From the theory of phase transitions, as described indetail by Bruce and Cowley (1981), comes the ratio

s

V0

kBTc ; 9

which specifies the type of phase transition.When s1 the phase transition is said to be of the

orderdisorder type. The local site potential is muchstronger than the thermal energies, so that distributionof orientations is sharply clustered around the minima ofthe local potentials. In general it is assumed that orien-tational ordering over distinct sites will give a behaviourthat is typical of orderdisorder transitions, which iswhy analogies are often made to pseudospin systems. In

the other extreme, s> 1, the phase transition is of thedisplacive type. Thermal energies are comparable to, orgreater than, the local site potential, so that there is awide distribution of orientations rather than sharp dis-tributions around the potential minima. In the presentcase we find that s3.5. Because of the relatively lowvalue of s, we might expect that the phase transition incalcite would share many characteristics with a typicaldisplacive phase transition, although it involves thedisorder of the carbonate groups. This is consistent withthe main features of the Fourier maps at high temper-atures, which certainly resemble more closely the dis-placive type of phase transition rather than a simpleorderdisorder picture.

There is one feature that is special to the case oforientational ordering and which is not found in typicaldisplacive phase transitions, namely that the local po-tential has rotational symmetry. This means that as the[001] librational amplitude (L33) increases, it willeventually become so large that the orientational prob-ability distribution function for the carbonate ions willencompass both the ordered orientation (h=0) and the

anti-ordered orientation (h=60). Then the carbonatesions will tend towards orientational disorder. In a waythis mechanism is a rotational analogue of Lindemannmelting. Unlike the pseudospin analogue, below Tc wewould not expect, in this case, to see a growing fractionof carbonate groups in the anti-ordered orientation.

It may be that calcite provides a new type of orien-tational orderdisorder phase transition. In most casesthe transitions in these systems are so discontinuous thatthe ordered phase is almost fully ordered at all temper-atures below the transition temperature. In such casesthere is little scope for discussion of the mechanism ofdisorder closer to the critical region, so it has often been

assumed that the transitions can be described within apseudospin formalism. We are suggesting that calciterepresents quite a different viewpoint. The observationthat in calcite the carbonate ion only occupies the oneorientational state in a manner becoming increasinglyless defined on heating, rather than flipping between twodistinct states, does make it difficult to see how anypseudospin model (e.g. Harris1999) physically maps tocalcite.

There are, however, some difficulties that must beconsidered. The first is that without a pseudospin for-malism it is not immediately clear how to define theorder parameter in microscopic terms. This is not to say

though that a quantity that reflects an order parameteron a macroscopic scale cannot easily be measuredtheexample of using the spontaneous strain to determinethe temperature dependence of the order parameter hasbeen described earlier. We suggest that the simplestdefinition in the present case is

Q 1 1

2 cos3#h i; 10

where the sign alternates between the layers along [001],since the carbonate groups in alternate layers are rotated

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by 60. In the limit Q fi 1, we can expand the cosine togive

Q 1 1

2 3#2D E

1 9

2L33: 11

However, 3# 2

D E is a fairly crude guide to the

ordering because the detailed shape of the orientationaldistribution function affects it. Also, because the tem-

perature factors are the quantities that are determinedwith least accuracy in a powder diffraction experimentand the TLS model is only really valid for smallamplitude motions, (11) is only at best approximated byour data. The macroscopic measurements of the tem-perature dependence ofQ are to be preferred over esti-mates taken from the measurements of L33. The mainresult, then, is not in the individual values of T and Ltensors, but their relative magnitude as a function oftemperature.

There are two important directions along which toconsider orientational correlations: intraplanar and in-terplanar. The existence of the R3c to R3m transition

signifies the loss of the latter, i.e. any average anti-cor-relation between carbonate layers. However, this disor-der could be achieved in several ways with regard to theintraplanar correlations. The symmetry change hasnothing to say about the disordering mechanism takingplace within a layer: the limiting cases would be indi-vidual free carbonate rotors or large-amplitude fluctua-tions of orientations within entire sheets, but could lieanywhere in between. Furthermore, diffraction and theextracted values of the TLS tensors cannot distinguishbetween individual free carbonate rotors or correlated,large-amplitude reorientations of groups of carbonateanions.

The local orientational potential is given by VLocal#,but we must now add the interactions between neigh-bouring carbonate groups. We restrict our discussion tonearest-neighbour interactions, and for the presentpurposes we aim to present the qualitative picture. InFig.7a we show a contour plot of the intraplanar ori-entational interaction between nearest neighbours. InFig.7b we show a contour plot of the interplanar ori-entational interaction between nearest neighbours.

The most striking point from Fig.7 is that theinteractions within a single plane are much stronger thanthose between the carbonates in different planes. Thismay imply that the loss of orientational ordering be-

tween neighbouring 001 layers takes place by correlatedL33 librations within each plane (i.e. each carbonategroup is far from being a free rotor), but with perhapslittle correlation between spins in neighbouring planes.Nevertheless, if each plane maintained absolute corre-lation of the orientations of all the carbonate ions, withthe disordering process being just a loss of correlationbetween the layers, it seems difficult to explain why therewould be an increasing amount of out-of-plane motionof ions, as revealed by the increase in the values ofT33andL11 with temperature.

It seems that even if individual orientations are notstrongly coupled between layers, whole layers do inter-act strongly, yielding the large anomalous strains up thec-axis, normal to the principal disordering processoccurring within 001. Lynden-Bell and Michel (1994)made the point that while in true molecular crystals,there are large interactions between molecular orienta-tions, in ionic compounds a lattice of counterions sep-arates the molecular ions, and the major interactionbetween orientations then becomes indirect. It is there-fore possible that the rotations and displacements of thecarbonates are coupled to those of the neighbouringlayers by the translations of the Ca ions. It has also beensuggested that translationrotation coupling may be themechanism by which Z and F-ordering are linked incalcite and sodium nitrate (Lynden-Bell et al. 1989;Lynden-Bell and Michel1994).

Summary

We have presented new diffraction data for calcite on

heating into the phase transition, which are consistentwith, but more extensive than, our previous measure-ments. Analysis of the crystal structure from the dif-fraction data shows that there is appreciable in-planethermal motion of the carbonate ions, which we modelby the L33 librational amplitude, and that this increasesrapidly on heating towards Tc. This suggests that themechanism of disordering is closer to that of a contin-uous rotational melting than that of a pseudospin dis-ordering. This agrees with the features observed in theorientational distribution functions of the simulations ofFerrario et al. (1994) and Liu et al. (2001). The anom-alous expansion of the c-axis, occurring normal to the

plane of the main disordering process, may be explainedby the growth of the amplitude of the out-of-planemotion. We have shown that the local single-particleordering potential is not significantly stronger than thenearest-neighbour orientational interactions. We suggestthat the transition in calcite is representative of a newtype of orientational orderdisorder phase transition.

Acknowledgements MTD wishes to thank EPSRC for financialsupport. We are pleased to acknowledge the collaboration withMark Hagen and Mark Harris. We would also like to thank RuthLynden-Bell for discussions on the molecular dynamics simulation.

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